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History of Classical Mechanics

Part II*, the 19th and 20th Centuries

C. Truesdell

Professor of Rational Mechanics, The Johns Hopkins University, Baltimore, Maryland 21218, U.S.A.

Continuing the story begun in Part I, this article sketches the development of rational mechanics in the nineteenth century. Then it was taught to every physicist and, as time went on, to an ever greater proportion of engineers. Not only was it the core of their training in natural science, but also it served as the paradigm of a scientific discipline. The rise of relativity and the theory of quanta left classical mechanics the preserve of engineers for the first half of the 20th century. Recently classical mechanics, particularly in reference to severely deformable bodies, has come to be studied and developed again as a science in its own right, much as it was in the Age of Reason and the Enlightenment.

Domination by the School of Paris

The Age of Laplace (c. 1770-c. 1825)

The work of Pierre Simon de Laplace (1749 1827) in pure mechanics concerns formally approximate applications rather than matters of principle. It is the kind of mathematics that can ultimately be replaced by the products of computing machines. Laplace is best remembered for his having derived crudely from Euler's general theory of fluids both approximate equations governing the motion of water in a shallow basin and an even coarser theory of the tides, for his mass of algebraic arithmetic regarding motions of the planets, and for his incredibly complicated ideas on capillarity. Laplace's real contribution was to the theory of heat. There he introduced the concept of a "sudden com- pression", today called an adiabatic process, and he suggested that the sonorous vibrations were of this kind. He calculated the relation between pressure and density in an ideal gas undergoing such a process. The resulting speed of propagation achieved what Laplace's predecessors for over a century had failed to do, namely, it explained and corrected the discrep- ancy between Newton's formula for that speed and the values obtained in experiments. Laplace upheld the idea that heat was an indestructible substance, but that was irrelevant to this particular result. The theory of conduction of heat in rigid bodies proposed and studied by Joseph Fourier (1768-1830) rests upon linear partial differential equations with linear

* Part I: Naturwissenschaften 63, 53 (1976).

boundary conditions and hence is amenable to infinitely many mathematical devices, most of which Fourier found and exploited. It, too, presumed that heat was an indestructible substance, so Fourier's attempts to extend it to deformable bodies were a failure. Several concepts of thermomechanics, for example the flux of heat, may be traced to Fourier's treatise, but in so primitive a form as scarcely to be discerned. On the one hand, Fourier' s theory isolated the conduction of heat as a phenomenon independent of forces and motions, which it generally is not. On the other hand, his work set forth clearly and definitely the simplest aspects of that phenomenon and thus challenged sub- sequent theorists to unite the theory of heat with the theory of force so as to describe the interaction of the two, and especially to determine the motive power of heat. Fourier systematically exploited expansions in terms of infinite series and integrals of trigonometric functions. He claimed to prove that such representations were valid for arbitrary functions, but his proofs were hopelessly wrong. Later mathematicians did establish theorems of the kind he claimed; still later mathematicians constructed a general theory of expansions in proper functions. Through application of these mathematical theories, Daniel Bernoulli's intuited principle of superimposed small oscillations was finally established as a theorem of mechanics.

The Age of Cauchy (c. 1820-c. 1850)

In order to calculate the resistance a fluid offers to shearing, it was necessary to conceive a theory of inter-

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nal friction. In order to account for the effects of torsion of elastic bars and the stiffness of elastic plates, it was necessary to conceive a general theory of shear stress in solids. Shear was the dominant concept of mechanics in the nineteenth century. The appropriate general theories were first constructed by Claude-Louis-Marie-Henri Navier (1785-1836). He represented a body as a static array of molecules; his

mathematics was questionable; and his results were subject to unnecessary restrictions. At the same time Augustin Jean Fresnel (1788-1827) in his attempts to explain double refraction by regarding the luminifer ous aether as an elastic substance went far toward con- structing a continuum theory of small elastic deformation. The subject was taken up at once by Augustin-Louis

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Cauchy (1789-1857), who broadened, deepened, and refounded it. He began by a penetrating analysis of the deformation of a continuum in general, irrespective of its constitutive properties. He introduced the general concepts of strain, local rotation, and stress. In order to do so he had to construct fairly general apparatus for handling tensors and tensor fields. He proved that every deformation might be regarded locally as the product of a pure strain by a rotation. Postulating that at a given point the traction was the same for all surfaces having a common tangent plane there, he proved the existence of a "stress tensor" which when applied to the normal to a plane element delivers the traction vector acting upon that element. Thus he united and broadly extended concepts and results of Euler and Coulomb regarding the forces the parts of a body exert upon each other. He applied Euler's principles of linear and rotational momentum to a continuous body and expressed them in terms of the stress tensor. The resulting equations are called "Cauchy's First and Second Laws of Motion" for a continuum. Cauchy reaffirmed the distinction, due to Jakob Ber- noulli and Euler, between general principles and constitutive relations. The constitutive relation of his theory of elasticity states that the stress tensor in an elastic body is alinear function of the infinitesimal strain tensor. This theory extends and simplifies Navier's. Cauchy accompanied his studies of an elastic continuum by counterparts for a regular array o r " lattice" of molecules. The two aspects of his work, the one atomistic and the other phenomenological, are re- flected by later developments leading, on the one hand, to the applied elasticity used by structural engineers and, on the other hand, to physicists' studies of the solid state. A molecular theory should reflect the influence of the kind of molecules assumed and hence should yield predictions more specific than those of the correspond- ing gross theory. Cauchy's molecular theory imposed relations reducing the greatest number of possibly independent elasticities of a general crystal from 36 to 15. These '' Cauchy relations" gave rise to much con- troversy. Later in the century they were shown to con- tradict the results of experiments on some materials. It became a major open problem to construct a molecular theory of solids that would be free of Cauchy's relations. Cauchy himself had suggested how it could be done: model the body as a lattice of two or more kinds of molecules, or as two or more interpenetrating lattices. His suggestion was not taken up until the twentieth century, and for a long time physicists believed that elasticity could not be given a satisfactory explanation by a theory of molecules. Cauchy was the first to see that the special symmetries shown by different materials could be represented by demanding that the constitutive relation enjoy invar- lance under a certain group of transformations. In particular, he introduced the concept of isotropy to represent response independent of direction, and he

determined the special forms that various kinds of functions must have if they are to be isotropic. If "Cauchy's relations" hold, an isotropic body has one and only one independent elasticity, not two. The conceptual structure which Cauchy created for general mechanics remained sufficient for at least a century after his death. Even more important and per- manent are numerous tools of algebra, geometry, and analysis he provided so as to promote the mechanics of deformable bodies. His influence in forming classical mechanics, as that discipline is nowadays understood and studied, is second only to Euler's. In fundamental usefulness, his concept of the stress tensor may be compared only with Newton's concept of force itself. Sim6on-Denis Poisson (1781-1840) developed the theories of elastic solids and viscous fluids in his own way, mainly on a molecular basis as far as fundamental theory was concerned. His study of elasticity, like much of Cauchy's, was directed toward the propagation of light, in the context of which it has lost its interest now, but his equations for longitudinal and transverse waves in an isotropic elastic body remain fundamental in acoustics. Earlier, in 1814, he had announced the partial differential equation for the vibration of plates; the problem had then been open for half a century, and various incorrect solutions had been published or circulated. Stimulated by Chladni's figures, the French Academy had offered a prize for the explanation of them; a young lady whose entry was totally incorrect was helped by the aged Lagrange, who recognized her great talent, as did also Fourier; Lagrange showed her the correct general equation in 1811, but he did not publish it, apparently donating it to the aspiring authoress. At her third attempt for the prize, in 1815, it was awarded to her. Her memoir contains Lagrange's equation but is otherwise dense with errors both major and minor. Were it not that the lady, if her portrait may be trusted, was decently endowed with feminine charm, the episode might serve as an early example of Women's Liberation. Poisson and others later obtained solutions of the fundamental equation, but in general they prescribed more conditions at the edges of the plate than can in fact be satisfied. Nothing specific regarding Chladni's figures was to be obtained from theory for a long time. Peter Gustav Lejeune Dirichlet (1805-1859), at one and the same time generalizing and correcting an argument of Lagrange, proved that a configuration which renders the potential energy of a discrete dynamical system a minimum was a dynamically stable one. Pois- son, William Rowan Hamilton (1805-1856), and Carl Gustav Jacob Jacobi (1840-1851) developed analytical dynamics in Lagrange's spirit as a branch of the formal theory of differential equations. This purely algebraic manipulation is the part of classical mechanics most familiar to physicists now. While Cauchy had constructed a magisterial analysis of all kinds of strain of bodies, his theory of elasticity was restricted to infinitesimal deformations from a


given, possibly stressed state. George Green (1793- 1841), even though he fixed as his object the theory of propagation of light in crystals, sought to remove any such limitation. At the same time, following the example of Lagrange, he posited a potential function, later called the stored-energy function. For elastic bands, such a function had been introduced a century earlier by Daniel Bernoulli, and its properties had been developed by Euler. Upon the three-dimensional theory it imposed a limitation. For example, in infinitesimal deformations a crystal might have as many as thirty-six independent elasticities according to Cauchy's continuum theory but only twenty-one according to Green's. Later developments favor the restrictions introduced by Green. However, Green's analysis, though just in idea, was not correct in detail for bodies subject to large strains. In an earlier work, devoted to electricity, Green had made a still greater contribution; namely, he had discovered and shown how to use the formula for converting the integral of the flux of a vector field over a closed surface into an integral over the region it included. This formula, sometimes called "Green's theorem" or "the divergence theorem", has been recognized as the most powerful tool in all of mathematical analysis, next to the infinitesimal calculus itself. It is now of everyday use in mechanics. The differential equations governing the flows of viscous and possibly compressible fluids, considered as continua, were derived by George Gabriel Stokes (1819-1903) in a memoir notable for its frankness as well as its clarity. These equations, which are called "the Navier-Stokes equations ", take their place alongside their special case, Euler's equation of ideal fluids, as defining the science of fluid mechanics in its tradi- tionally limited sense. As had Cauchy earlier, Stokes prefixed to his constitutive considerations a thorough and illuminating kinematic analysis, in which he revealed the nature of the vorticity as being a local angular velocity. Regarding the fluid as a continuum, he appealed to a concept of internal friction which generalizes Newton's. On the basis of his results he corrected Newton's analysis of the flow induced by a spinning cylinder, but its comrade, the flow around a spinning sphere, remains to this day uncorrected because still undetermined within Stokes' general theory. The Navier-Stokes equations were designed so as to take account of the resistance offered by fluids which adhere to bodies that move in them, and especially so as to avoid the" d'Alembert paradox". Stokes was unable to calculate the required solutions except by dubious approximations. The partial differential equations of his theory are notoriously difficult to use in cases appropriate to natural problems, and study of their properties affords a major domain of research in mechanics today. Stokes provided also an analysis of the absorption of sound in a fluid, but it is incomplete because the only dissipative mechanism he considered was viscosity. He was not in a position to take conduction of

heat into account, because the connection between heat and work had not yet been found. This was to .be the province of a new science called thermodynamics.

The Main Disciplines of Mechanics

Thermodynamics

Nicolas Lhonard Sadi Carnot (1796-1832) asserted that the maximum motive power a given quantitiy of heat could produce in falling from one temperature to another was obtained by absorption at the higher temperature, followed by adiabatic cooling, followed by emission at the lower temperature, followed by an adiabatic return to the higher temperature. The process so described is a "Carnot cycle", and a body executing it is called an "ideal Carnot engine". Regarding heat as an indestructible substance, from his general assertion Carnot derived conditions relat- ing the specific heats of the working body to its motive power as a Carnot engine. By comparing the specific heats with measured values, he hoped to infer the value of the motive power. The working bodies Carnot considered were ideal gases. Carnot's theory implied that at least one of the specific heats of an ideal gas had to be a decreasing function of the volume. This result could not square with experiment. As time went by, James Prescott Joule (1818-1889) became convinced that heat and work were universally and uniformly interconvertible, and he devised a sequence of fine experiments to support his conclu- sion. William John Macquorn Rankine (1820-1872), Rudolf Julius Emmanuel Clausius (1822 1888), and, a little afterward, William Thomson (1824-1907), who later became Lord Kelvin, accepted the new idea of heat and thus had to reject Carnot's theory. Rankine's analysis was intertwined with his picture of bodies as assemblies of vortices. Clausius regarded bodies as being assemblies of moving mass-points or balls, but he developed a phenomenal theory of heat and work, independent of his ideas about molecular structure. This theory came to be called thermodynamics. According to it, a given amount of work is produced only at the expense of a certain amount of heat. A body has a certain internal energy, determined by its volume and temperature, and in a cyclic process it is this energy, not the heat, that is conserved. This is Clausius' First Law of Thermodynamics. Clausius' theory is free of the difficulties inherent in Carnot's. As Thomson showed, it determines the effi- ciency of a Carnot cycle explicitly, and it is consistent both with the Laplace-Poisson law of adiabatic change and with two other simple properties of ideal gases: the first, proposed in effect by Julius Robert Mayer (1814-1878), that the difference of the specific heats of an ideal gas is a constant, and the second, proposed explicitly by Karl Heinrich Alexander Holtzmann (1811-1865), that the latent heat of an ideal gas is

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the mechanical equivalent of its pressure. In fact Clau- sius adopted Holtzmann's relation and made it one of his assumptions. In later work Clausius discovered a second important function of the volume and temperature of a gas. This function, which he subsequently called the "entropy", is the value of the integral of the heating divided by the temperature. That that integral is the value of such a function is the Second Law of Thermodynamics according to Clausius. The energy function and the entropy function of a gas determine all its thermodyna- mic properties, and the whole doctrine of thermodynamics, in its early form, is expressed compactly by the two laws. The theory so far refers to processes which if reversed will release exactly the heat gained and the work done. That this is so, follows from the underlying classical doctrine of latent and specific heats, and from the assumed thermal equation, which makes the pressure of a gas a function of its volume and temperature. This class of constitutive relations was employed without question by all the early thermodynamicists. In nature, neither heat nor work is totally recoverable if a process is reversed. The conduction of heat, the friction that opposes motion, and the diffusion of mass in a mixture are examples of irreversible processes. To include them within a theory of thermodynamics, it is necessary to replace the classical constitutive relations for heating and working. Fourier's theory of the conduction of heat in rigid bodies might have been invoked as a starting point toward a general theory of the relations between heat and work. Instead of doing such a thing, Clausius laid down a restriction on the outcomes of irreversible processes. They must result in an increase of entropy at least as great as the integral of the heating divided by the temperature. According to Clausius, this is the Second Law for possibly irreversible processes. Thom- son claimed that Clausius' Second Law was equivalent to a prohibition of what was later called perpetual motion of the second kind, namely, a cyclic process that does positive work without emitting any heat. It is impossible to use this strengthened Second Law as the basis of rational analysis, since, unlike the laws for reversible processes, it implies use of unstated constitutive relations about which we know nothing except what they are not. Thermodynamics, instead of pro- gressing so as to include explicit and definite constitutive relations such as to represent bodies that may undergo irreversible processes, retreated into a maze of denials. With no change at all in its mathematical structure, no new theorems, and hence no progress, it came to be regarded as the science of infinitely slow or ~ processes, in which a body changes its state so slowly as to remain always in equilibrium. Mathematics is of no help here, for it cannot deal with self-contradictory statements, no matter how appealing they may be on grounds of physical intuition. The thermodynamics of irreversible processes remained stagnant for a century.

The thermodynamics of reversible processes-more accurately, the thermodynamics of materials with constitutive relations such as to render all processes reversible-was broadened so as to include materials such as elastic solids and mixtures of fluids, and it was applied successfully to explain many aspects of chemi- cal equilibrium. Josiah Willard Gibbs (1839-1903) saw that the formu- lae of the thermodynamics of reversible processes could be interpreted in variational terms. He constructed a pure thermostatics, though unfortunately he chose to call it, in imitation of the earlier theory with which it was parallel, thermodynamics. In his theory there are no processes, and equilibrium is defined by comparison of infinitely many putative conditions of a given body. Thus, necessarily, Gibbs considered fields defined in space, in contrast with the functions of time alone that entered the theories of Carnot and Clausius. In the class &all fields of specific volume and temperature that correspond to given total volume and energy, Gibbs defined equilibrium as being the fields that rendered the entropy a maximum. He proved that these fields were uniform and were con- nected to each other and to the energy and entropy by relations having just the same form as their counterparts in Clausius' theory of reversible processes. Furthermore, Gibb's definition implies further restrictions beyond those of Clausius' theory. These we may describe as ensuring the stability of equilibrium. Most of Gibbs's work refers to mixtures, not just to a single substance. Gibbs maintained a high standard of clarity and mathematical rigor, but he presented his reasoning mainly in words, and his work has been widely misunderstood. Expositions of it usually misrepresent it as being a kind of thermodynamics of quasistatic processes. As thermostatics, which it is, it is perfect within its inherent limitations. It says nothing at all about bodies not in equilibrium. In particular, it affords no frame- work upon which we could formulate the question of whether a body constitutiveb7 incapable of reversible processes can ever attain equilibrium. Its role in explanation of thermal phenomena is precisely ana- logous to that of the statics of forces, which restricts mechanical equilibrium but gives no information at all about motions of bodies which are not in equilibrium.

Elasticity in the Second Half of the Nineteenth Century

As was to be expected, Green's definitive theory of small elastic strain was cultivated intensely and abundantly. Three researches deserve notice because they probe matters of principle. The first two of them con- cern torsion of bars, but in very different ways. The Bernoulli-Euler theory of elastic bands represented the rod as a plane curve susceptible of bending alone, and in its own plane. Coulomb's experiments referred to torsion alone. The general theory of elasticity subsumes both phenomena in principle, provided the displacements and strains be small enough, and


it should serve to interrelate them, but how it did so was far from obvious. There were two reasonable courses a theorist might follow. First, using the three- dimensional theory as a basis, he might associate with a curve some measure of torsional stiffness as well as the Bernoulli-Euler modulus of flexural stiffness. Second, he might abandon the theory of one-dimensional continua altogether and consider a rod as a special kind of body in three dimensions. Gustav Robert Kirchhoff (1824-1887) chose the first course. He considered a rod as a possibly skew curve, with each point of which was associated a cross-section. By performing an approximate integration of the equations of the three-dimensional theory over that cross-section he saw how to replace the effect of its shape by a scalar modulus associated with vectors normal to the curve. The relative rotation of those vectors represents the twist of the rod, and the modulus is the torsional stiffness. In this way Kirchhoff was able to find ordinary differential equations such as to describe simultaneous bending and twisting. This theory is of importance far greaLer than might seem, for it provides a pilot case for the concept of a body composed not only of points but also of vectors associated with those points- that is, a structured or oriented material. The second course was chosen by Adh6mar J.-C. Barr~ de Saint-Venant (1797-1886). Considering first a bar subject to torques acting upon its plane ends, he recognized that infinitely many different arrangements of the shearing stresses on those ends would give rise to those same torques, yet according to the three-dimensional theory each of those different arrangements would produce a different deformation of the bar and different stresses within it. When a real bar is twisted, we cannot know the details of the way the grips act upon it. Moreover, experience shows that these details make no difference, at least for long bars. There must be some property common to all the infinitely many solutions of the problem, a common property far more important than the differences among them. Indeed, Coulomb's experiments had shown that the torque was proportional to the twist it produced, and the theory of elasticity ought to determine the constant of proportionality that corresponds to a given cross- section. This constant should be at least approximately the same for all the infinitely many solutions the theory allows for the torsion of a given bar. This was a problem of an altogether new kind. Up to this time theories of greater and greater refinement had been found so as to represent natural phenomena in more and more detail. Here the detail appeared largely superfluous. What was called for was a way to cut through the detail so as to get to the essential information that the theory could provide. Saint-Venant conjectured that if each end of the twisted beam were subjected to an additional set of loads equipollent to null, the differences of stress and strain effected thereby would be small, at least at dis- tances far from the ends. This assertion, while plausible

and widely accepted pragmatically, remains unproved to this day. It is called "Saint-Venant's Principle". If Saint-Venant's Principle is true, then any one of the infinitely many solutions of the problem of torsion will suffice to determine the torsional modulus and other major features. Saint-Venant himself chose the solution which seems simplest, namely, that in which each cross-section of the beam, although warped, retains its contour, which is rotated through an angle proportional to its distance from one end. With admir- able clarity and directness, Saint-Venant proved that there was one and only one solution of this special kind; he showed how to obtain it in general; he calculated all the details for simple but important cases; and he obtained a simple, general expression for the torsional stiffness of any contour. Saint-Venant treated the more difficult problem of flexure with the same approach and the same measure of success and completeness. For penetration of concept, for assessment of a key problem, and for skill in solving it, his work stands as one of the supreme monuments of mechanics. Taking up the theory of plates, Kirchhoffgave reasons for replacing Poisson's two boundary conditions by one. He was thus able to obtain important special solutions for circular plates and to explain some of Chladni's figures. To this day, however, it cannot be said that all of Chladni's results are accounted for by the theory. They stand as an example, very rare, of how experiment can stimulate imporant discoveries in theory yet not be superseded by it. Kirchhoff's ingenious method for causing Poisson's two natural conditions at the edges to coalesce excited great attention and has continued to do so. It is the earliest example of what is now called a "boundary-layer effect" in the relation between a general or "exact" theory and an "approximate" theory designed to obtain average values over a class of solutions according to the general one. Like Saint-Venant's principle, it furnishes even today an important subject of research. In the theory of large elastic deformations little progress was made. The correct general equations were found in principle by Kirchhoff and explicitly by Kel- vin; Joseph Finger (1841-1925) worked out the details for isotropic materials; but no-one solved any illuminating special cases or proved any characterizing theorems. The molecular theory of elasticity, laboring under the defect o f " Cauchy's relations ", was not studied much. Jules Henri Poincar~ (1854-1912) showed that the strain energy, considered to be the sum of the pair potentials of the atoms, was not strictly additive, and he estimated the error that resulted from assuming it was SO.

Fluid Mechanics in the Second Half of the Nineteenth Century

Immediately after the formulation of thermodynamics Kirchhoff saw that to estimate the absorption and

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dispersion of sound waves in fluids it was necessary to take into account not only viscosity but also the conduction of heat. In order to establish differential equations of motion, he had to express the principle of conservation of energy as a field equation, on a par with Cauchy's Laws of Motion. He did so by sup- posing that the fluid obeye d locally a caloric equation of state like that which Clausius had found for homogeneous conditions. Although Kirchhoff considered only small motions of an ideal gas, his concepts and methods are general, and his work stands as the first attempt, and a successful one, to treat continuum mechanics and thermodynamics as a single, unified discipline. Mechanical and thermal effects are insepar- ably coupled in his theory, as they ought always to be except in very particular instances. The kind of results to expect here had been suggested by work of Maxwell on the kinetic theory of gases (see below). For his final results Kirchhoff supposed the frequency of the sound waves to be small and so found dispersion negligible but absorption proportional to the square of the frequency, with a coefficient which is a linear combination of the shear viscosity, the bulk viscosity, and the heat conductivity. The "vort ici ty" or local angular velocity of a fluid motion had appeared in the equations of hydrodynamics since the beginning but had been little understood. Much of the early studies had assumed, just so as to attain a tractable case, that it was null, and an extensive theory of such " i r rotat ional" flows had been developed. The meaning of the vorticity had first been explained kinematically by the analyses of Cauchy and Stokes. Hermann Ludwig Ferdinand v. Helmholtz (1821-1894) saw that the behavior of the vorticity in a flow of an Eulerian fluid could be described precisely even though the differential equations of motion could only rarely be solved fully. He introduced the trajectories of the vorticity field, which he called vortex lines. The particles of fluid spin about these curves as axes. Helmholtz proved that the fluid particles which at one time constitute such a curve do so forever, and that the flux of the vorticity field through any surface swept out by a set of fluid particles as they move remains constant. In a portion of an Eulerian fluid the vorticity thus enjoys a certain per- manence. It can never be created or destroyed; in particular, a portion of fluid once in irrotational flow has always been so and always will be; and if the vorticity is not null, its magnitude waxes or wanes accordingly as the vortex lines come closer together or spread apart. As Helmholtz said, his theorems make the flows of fluids "approachable in concept" even when they are too difficult to determine in detail. Kelvin defined the circulation around a circuit of fluid particles as the total tangential component of velocity of that circuit. Thus it is a measure of the circuit's mean speed of rotation. Kelvin proved that as the particles of an Eulerian fluid that make up a circuit at any one time move in the course of time, all the circuits they have occupied or come to occupy have

the same circulation. He showed that Helmholtz's theorems are easy consequences of this fact. Kelvin's theorem is often called the fundamental theorem of classical hydrodynamics. It is equivalent to a simple explicit formula that Cauchy had published in one of his early papers, but Cauchy's achievement had not been fully explained by him or understood by others. The researches of Helmholtz and Kelvin serve as examples of a new approach to problems of mechanics. Recognizing that the non-linearity of the hydrodynam- ical equations made them insusceptible of wholesale explicit solution along the lines of the linear theories of elasticity and the conduction of heat, Helmholtz and Kelvin demonstrated properties of all solutions without determining any one solution. This kind of analysis, which, although it is strictly precise, is called "qualitative", has become more and more fruitful as techniques of analysis have been perfected. Hydrodynamics has provided most of the key examples of principles and methods in mechanics. Its non- linearity makes a frontal attack extremely difficult, though not necessarily impossible. Indeed, Georg Bernhard Riemann (1826-1866) created a general method for calculating such gas flows as depend upon only two independent variables. There are five other approaches to hydrodynamics. One is to replace the exact equations by linear ones designed to approximate a particular situation. Key examples are furnished by acoustics: Euler's definitive treatment of infinitesimal waves in ideal fluids, and Kirchhoft~s analysis of plane oscillations in a viscous fluid. Another is to isolate a class of solutions in which the non-linear terms can somehow be evaluated directly. Here the key example is Euler's theory ofirrotational flows of an incompressible fluid, in which the velocity field is determinate by itself, and the pressure field can be calculated explicitly in terms of the velocity. The third is qualitative analysis, used by Helmholtz and Kelvin to classify and describe the flows that are not irrotational. The fourth is by ingenious transformations to convert the non-linear differential equations into linear ones, possibly at the expense of making the boundary conditions more complicated. Here a fine example was provided by Sergei Alekseevich Chaplygin (1869-1942), who showed how problems of steady gas jets could be solved if velocity rather than position were taken as the independent variable. The fifth, the method of singular surfaces, will be described now.

New Currents in the Latter 19th Century

Pierre Henri Hugoniot (1815-1887)

A disturbance may be regarded as small for two different reasons : it may be of small magnitude, or it may be confined to a small region. The tradition of acoustics chose the former possibility and for the most part still does. If applied to small disturbances, the general

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and typically non-linear partial differential equations of a field theory may be replaced, it seems, by linear ones, easy to solve by use of suitable harmonic oscillations. The process of approximation, while it may or may not be found just in the end, altogether excludes the typical phenomena of the exact theory, namely, decay or reinforcement of a wave as it progresses. Hugoniot chose to exploit the second possibility. He took the disturbance as being a discontinuity of arbitrarily large amount but localized to a single surface. In special cases Euler had considered discontinuities of the velocity and its gradient, but only in linear acoustics; Stokes and Rankine had fleetingly considered normal discontinuities in the velocity, which are called shock waves, but later Stokes had allowed himself to be deceived by a fallacious argument presented first by Kelvin and then by John William Strutt, Baron Rayleigh (1842-1919), and had retracted his conclu- sions; and Helmholtz had considered tangential discontinuities, called vortex sheets. Hugoniot united and extended all this work in a general theory of singular surfaces. He derived conditions of compatibility to express the assumption that the discontinuity was spread out smoothly over the surface and that it per- sisted in time. By combining these with the principle of momentum he showed how to determine the pos- sible directions of the amplitude of the discontinuity and the speed with which a wave of given amplitude might propagate. Hugoniot applied his methods mainly to discontinuities of the acceleration, which he identified with sound waves. The bodies he considered were Eulerian fluids, finitely elastic solids in plane motion, and infinitesimally elastic solids in general motion. Hugoniot's work might have fallen into oblivion had not Pierre Duhem (1861-1916) taken it up. He explained it to Jacques Hadamard (1865-1963), who developed it into a general theory and wrote a cele- brated treatise about it: Lefons sur la Propagation des Ondes. His theorems show that while weak waves leave the vorticity field unaffected, passage of a curved and oblique shock wave generates vorticity. He calculated the condition for the amplitudes of weak waves in a finitely strained elastic body and demonstrated con- nections between wave propagation and the stability of equilibrium. Duhem showed that waves in Hugoniot's sense are generally impossible in linearly viscous bodies, and he suggested that in such bodies extremely rapid though smooth changes could occur in thin layers. From development of this concept grew the modern theory of shock structure. The methods and concepts of Hugoniot are general. They are applied more and more frequently to study the way disturbances are propagated in bodies of the complicated materials considered in mechanics today.

James Clerk Maxwell (1831-1879) and his Influence

Simple theories which represent a gas as a vast assem- bly of molecules flying hither and thither at random

had been proposed by Daniel Bernoulli and John Her- apath (1790-1869). This elementary kinetic theory had been worked out admirably by John James Waterston (1811 1883), but his memoir was denied publication because the British physicists of the day found re- pugnant a molecular hypothesis mathematically expressed. Clausius, writing fifteen years later, em- braced the kinetic view of matter and introduced an explicit probability so as to calculate the mean free path of a molecule. Maxwell thereupon took up the subject. Introducing the molecular density function, that is, the function whose integral over a region of space and velocities gives the expected number of molecules there at a given time, he determined the form it must have if it were to depend upon the molecular speed alone and to render all directions of motion equally probable. This is the "Maxwellian density", namely, an exponential function of the square of the random speed of the molecule. Maxwell showed this density function to be compatible with what should be expected of a gas in equilibrium. In his second paper on the subject Maxwell started afresh and laid out plausible assumptions regarding the rate of increase of the molecular density as a result of collisions among the molecules. His basic idea, as later recast by Ludwig Boltzmann (1844-1906), is embodied in the Maxwell-Boltzmann integrodiffer- ential equation. This equation of evolution defines the classical kinetic theory of moderately rarefied monatomic gases. The temperature, stress tensor, heating flux vector, etc. are calculated from the molecular density function, and the gross behavior of the gas is determined, if only the equation can be solved. To solve it precisely, however, or to survey the nature of its solutions, seems to be the most difficult problem in all of mechanics. Maxwell showed that for a special molecular model, chosen for no other reason than mathematical tractability, the gas behaved in first gross approximation like a special case of the continuum fluid of Stokes, endowed with the power to conduct heat according to Fourier's law. Maxwell calculated the viscosity and heat conductivity explicitly; he found both to be proportional to the temperature. He estimated their relative importance as expressed by their ratio multiplied by a specific heat; he found this pure number to be about 1. The discovery of thermal transpiration by Osborne Reynolds (1842-1912) suggested that the Navier- Stokes and Fourier constitutive relations for viscous, heat-conducting fluids were insufficient to describe flows of rarefied gases. Maxwell devoted his third paper on the kinetic theory to an attempt to correct those relations by more careful development of his earlier ideas. His ingenious approximate calculation delivers the stress tensor as a non-linear function not only of the velocity gradient but also of the temperature gradient. Although his attempt to explain thermal transpiration was not satisfactory, Maxwell's results are of deep importance for the light they cast toward the future of mechanics. They suggest that between


mechanics and thermodynamics there is no separation in principle. Maxwell's work proves untenable the pre- viously traditional idea that differences of displace- ment or velocity alone can produce internal forces and that differences of temperature alone can make heat flow. Such a separation corresponds to no more than a particularly simple constitutive assumption, not a general law of mechanics. From this time onward critical students of the foundations saw that mechanics and thermodynamics had to become a single, unified theory. Boltzmann followed and extended Maxwell's ideas, sometimes happily and sometimes not. He attempted to prove that Maxwell's distribution, suitably genera- lized, was appropriate to statistical equilibrium of any numerous dynamical system and therefore provided a description not only for monatomic gases but also for polyatomic ones and for liquids and solids. The result is called "Boltzmann's Law". Boltzmann supported his claim by heuristic reasoning that is not always correct, and in the next century his theorem, after due amendment, was finally proved to be valid as an asymptotic formula in the limit as the number of degrees of freedom is allowed to approach in- finity. Boltzmann took the probable value of a function of the configuration as being its average over a very long time, and he identified probable values with what gross experiments can measure. One of Boltzmann's arguments rests on the suggestion that a dynamical system ultimately occupies every configuration consistent with its assigned total energy. It is easy to see that this "ergodic hypothesis" cannot be true of smooth motions. Maxwell suggested that collisions among molecules could introduce discontinuities such as to make a dynamical system behave this way, but nothing of the sort has ever been proved. In his fourth and last paper on the subject of statistical mechanics he greatly improved Boltzmann's treatment by introducing "phase space", the space of the co- ordinates and momenta used in Hamilton's formulation of analytical dynamics. Maxwell laid down as basic the assumption that any probability assigned to a set of phases at some one time must be convected by the motion. This is the "principle of conservation of density in phase ". Maxwell's averages are averages in phase space at a fixed time, not averages in time along a particular motion. For the case of an ideal gas Maxwell showed how to calculate phase averages explicitly. If the ergodic hypothesis were true, the phase average of any given function would equal its time average on almost any trajectory, so Maxwell's phase averages would deliver what Boltzmann sought. The ergodic hypothesis is false, but the results Maxwell and Boltzmann inferred from it may be true in some sense of approximation or probability. To show that indeed they are, with important reservations, was left for mathematicians of the twentieth century. Boltzmann attempted to reconcile the dissipationless, reversible motion of a Newtonian dynamical system with the fact that dissipation is an everyday occurrence

in nature. Natural bodies tend, if left to themselves, to settle into equilibrium, never to depart from it as time goes on. Boltzmann claimed that somehow motions going in one sense were "more probable" than those going in another. His point of view came later to be shared, more or less, by physicists, although mathematicians rejected his arguments. Some of the "paradoxes" that were then discovered and debated are famous. One of them rests upon a great theorem proved by Poincar6: A dynamical system confined to a finite region returns arbitrarily near to almost every phase it ever occupies. Therefore, no continuous function of the phase of any such system can ever show any trend in time. Objections of this kind do not touch the theory of statistical equilibrium, but they show that the kinetic theory of gas flows and other statistical theories that imply a preferred trend in time cannot be strict consequences of analytical dynamics for any single system. They leave open the possibility of asymptotic comparisons over a sequence of ever more numerous systems. Gibbs outlined a postulational approach to statistical mechanics. Laying down the Maxwell-Boltzmann density of probability, which he called "canonical", as a particular case worthy of interest, he showed that from it could be derived relations formally identical with those of thermostatics. The approach of Gibbs, which sets aside any attempt to relate statistical mechanics with analytical dynamics, lends itself to applications and is widely used by chemists today.

Mathematical Theories of Stability

I cannot describe here the growth of parts of mathematical analysis that bear upon mechanics. One of these, however, must be mentioned: the theory of dynamic stability. It arose in mechanical contexts such as the orbits of the planets and the oscillations of spinning masses of fluid. Prototypes were provided by Poin- car6's analysis of systems with two independent variables and discovery of conditions sufficient for perio- dic motions. The most powerful methods were introduced by Aleksandr Mikhailovich Lyapunov (t85% 1918). These have been applied and extended to ever more numerous and difficult special problems of mechanics, especially in the last three decades. They join Hugoniot's methods to provide the most powerful tools at the disposal of the theorist of mechanics today.

The Decline of Classical Mechanics.

The Effects of Relativity and Quantum Mechanics upon Classical Mechanics. The Beginnings of Applied Mechanics

The word "classical" has two senses in scientific writing; (1) acknowledged as being of the first rank or authority, and (2) known, elementary, and exhausted


("trivial" in the root meaning of that word). In the twentieth century mechanics based upon the principles and concepts used up to 1900 acquired the adjective "classical" in its second and pejorative sense, largely because of the rise of quantum mechanics and relativity. "Fundamental" in physics came to mean "concerning extremely high velocities, extremely small sizes, or both". Physicists gave less and less attention to classical mechanics because they thought nothing more could be learned from it and nothing new discovered about it, although of course they continued to use it in the design of the exPerimental apparatus with which they claimed to controvert it. At about the same time "applied" in mathematics came to refer not to the object studied but to the originality and logical standards of the student, again in a pejorative sense. Engineers still had to be taught classical mechanics, because in terms of it they could understand the machines with which they worked and could devise new machines for new purposes. Research in mechanics came to be slanted toward the needs of engineers and to be carried out largely by university teachers who regarded mathematics as a scullery-maid, not a goddess or even a mistress. Leading exponents of applied mechanics were Ludwig Prandtl (1875 1953) and Geoffrey Ingham Taylor (born 1886). Prandtl, noticing that the flow of a real fluid past a body was often very much like that of an Eulerian fluid except in a thin zone alongside the body, introduced the concept of a " boundary layer" which represented the effects of viscosity. He obtained differential equations he regarded as sufficient to approximate the flow in this layer; his discovery has been inestima- bly useful to engineers, and an enormous literature is grown up regarding it. In assessing Prandtl's place in history we may observe that while for its practical usefulness the boundary layer can be compared with Saint-Venant's principle in elasticity, the two are fun- damentally different. Saint-Venant perceived a method of setting aside superfluous information obtained when a general theory is applied to a certain class of problems; no mere computation could ever dispense with his concept and his achievement. The same may be said of Kirchhoff's theory of the vibrations of thin plates, for details of the way a plate is supported usually cannot be known and hence must be unimportant in the theory. Prandtl's boundary-layer theory provides a method of approximate calculation, neither more nor less; far from eliminating inascertainable information, it neglects an important and natural boundary condition, and to compare its predictions with experiment is a touchy matter, more art [han science. In principle, all results it is designed to yield could be gotten more accurately by solving the Navier- Stokes equations. At such time as techniques of computation will have been perfected sufficiently to enable engineers to apply the Navier-Stokes theory correctly to flows past obstacles, boundary-layer theory will have been superannuated.

Observations going back to Leonardo's time show that the flows of fluids most commonly observed are tumul- tuous and complex rather than smooth and simple. To calculate such flows or otherwise understand the nature of turbulence, has been recognized for at least a century as a major open problem of mechanics, perhaps the most important of all. Notable and plausible early work by Joseph Boussinesq (1842-1929) and by Reynolds failed to get far with the problem or to inspire their successors to create a good theory. Although some of the greatest experts as well as multitudes of employees have studied turbulence and are still study- ing it, with boundless expense of money and time, it would be hard to find in all this work any achievement sufficiently definite as to deserve a place in a short summary of the main course of mechanics. At this date, turbulence seems still to be a craft of experiment. The history of classical mechanics is a history of key special problems. One such problem has provided the statement of the next, more difficult or more complicated, and often has opened the way to it. The great inclusive theories have grown inductively from the suc- cesses of narrower theories, stimulated once in a while by experimental discovery of a new phenomenon. The applied mechanics of the first half of the twentieth century was different in kind. Stressing applications at the expense of principle, it accumulated mountains of special cases, indeed "applications", stored like lumber against the possibility that some day some engineer might use them. There is a second difference. Down to the end of the last century mechanics made use of the latest discoveries of mathematics, and indeed many of those discoveries were motivated by problems arising in mechanics. The mechanics labelled applied for the most part turned its back upon contemporary research in mathematics. Not only were the disciplines of mechanics established before 1900 regarded as final, but useful discoveries in mathematics seemed to end at about the same time, too. Applied mechanics was indeed "applied" and indeed "classical ".

The Twentieth Century

Fundamental "Classical" Mechanics in the Early Twentieth Century

Not all researches in mechanics in the early years of this century belonged to the category of "applied mechanics ". Duhem, whose work did not attract until recently the notice it deserves, made notable but only partly successful efforts to unite mechanics and thermodynamics. Georg Hamel (1887-1954) proposed axioms of mechanics in which the deformable body was the typical object of interest and forces were to some extent brought out of the shadows of intuition and made elements of a mathematical system. His was an attempt to render explicit, general, and precise the approach of Newton and Euler, but he did not succeed entirely.


Max Born (1882-1970) solved the problem, left open since the days of Cauchy and Poisson, of finding an adequate molecular model for an elastic body. Sub- stantiating a suggestion left by Cauchy himself, Born showed that a multiatomic or interpenetrating lattice was not necessarily subject to the "Cauchy relations" and thus could represent a general crystal. Aside from studies which, important as they were, pertained more to analysis than to mechanics, the most interesting work in mechanics in the first half of this century was done in the kinetic theory of gases. David Hilbert (1862 1943) set up a formal method for calculating particular solutions of the Maxwell-Boltzmann equation. These solutions are general enough to allow for all fields of density, temperature, and velocity, but each of them is only one of the infinitely many solutions that can correspond to a given set of such fields. Hilbert's procedure, though purely formal, indicates that such solutions should exist and shows how to calculate them in principle by successive approximations. The concept reminds us of Saint-Venant's principle in elasticity in that a vast amount of unimportant detail about the general solutions is set aside inten- tionally in Hilbert's search for the particular solutions believed to be typical in some sense and also the most important. There is a great difference, however: Saint- Venant worked with an underlying theory already rather well understood, a linear theory in which it is easy to obtain solutions of specific problems; on the contrary, it was not known, and still it is not known in sufficient generality, whether the Maxwell-Boltz- mann equation of the kinetic theory has solutions in general or whether the special class indicated by Hil- bert exists. Sydney Chapman (1888-1970) and David Enskog (188J~1947), presuming implicitly that there were solutions of Hilbert's type, concocted prescrip- tions for working toward them directly. At the first stage the constitutive equations of Eulerian hydrodynamics result, and at the second stage, the Navier- Stokes and Fourier equations. The viscosity and heat conductivity are determined explicitly, thus solving a problem set by Maxwell and labored upon in vain by Boltzmann. Higher approximations, generalizing Maxwell's theory of rarefied gases, can be calculated by the same procedure. It remains to this day a challenge to the student of the principles of mechanics to prove that solutions of a type that may be compared with those of fluid dynamics really do exist, and to determine their position in regard to the infinitely more numerous solutions that may correspond to the same fields of gross flow.

Some Recent Trends in Rational Mechanics

A renascence of classical mechanics as an independent science began in the late 1940's. Here I can do no more than point out some of its trends. 1. The old open problems concerning Eulerian and Navier-Stokes fluids are taken seriously again. Many

of them have been solved with elegance and finality by use of new mathematics. Examples may be found in gas flows with Shock waves, flows of incompressible fluids with free boundaries, flows containing vortex rings, concepts and methods in hydrodynamic stability, swirling flows and tornados in viscous fluids. 2. Solutions to key problems of finite elastic deformation have been discovered. The qualitative theory has been developed through general theorems, and specific properties of shock waves and acceleration waves in elastic solids have been determined. 3. Maxwell's kinetic theory of gases has been shown to yield definite predictions about certain gas flows, not approximately in the sense of nineteenth-century hydrodynamics but as flows of fluids of a general kind. 4. The ergodic theory of George David Birkhoff (1884- 1944) and others has been shown to provide a satisfactory basis for the statistical mechanics of equilibrium. Also a correct treatment of the equilibrium of polyatomic fluids according to statistical mechanics has at least been constructed. 5. Partly as a response to the invention of various new materials, but at least equally because of intrinsic interest, new constitutive relations have been introduced and studied. A general theory of constitutive relations has been formulated so as to provide criteria that such relations should satisfy and to interrelate classes of materials. A key example is the "simple fluid ", a material which may have long-range memory but no preferred configurations. This fluid serves to model the behavior of high-polymer solutions. Anal- yses based upon it have successfully predicted that a body of such fluid will climb up a rod rotating within it, will swell upon emergence from a tube, will bulge if allowed to flow downward in a trough, etc., all of these phenomena being outside the range of nineteenth-century hydrodynamics. 6. The concept of oriented material, of which the proto- type is the Bernoulli-Euler-Kirchhoff theory of elastic rods, has been extended so as to model various sub- stances of current interest, especially liquid crystals. The response of a liquid crystal to shearing depends upon the direction in which the shear is effected. 7. The concept of force has been made formal through a set of axioms, much as points and lines are made formal in geometry and mass is made formal in measure theory. Systems of forces are undefined objects, subject to explicit rules of operation. Euler's laws of mechanics can be replaced by a single, simple axiom: The power of a system of forces is the same for all observers. Newton's law of action and reaction, if prop- erly understood, is a corollary of the new axiom. The existence of the traction vector and Cauchy's pos- tulate about it are also consequences of the general theory of forces and the assumption that contact forces depend only upon the common boundaries of bodies and vanish with the area of such a boundary. 8. The Second Law of Thermodynamics has been formulated in generality sufficient to apply to deformable bodies undergoing processes of any kind. It is to be

Naturwissenschaften 63, 119- t30 (1976) :D by Springer-Verlag 1976 129

understood as a restriction upon constitutive relations in the sense that they should be identically compatible with it for any process whatever. 9. A general thermomechanics of materials with memory has been constructed. Within it, the extremal principles which had to remain outside the formal structure of nineteenth-century thermodynamics have been proved as theorems. For example, if a particle of such a body has always been at rest and then is subjected to a cyclic process, its free energy afterward cannot be greater than if it had continued to remain at rest. 10. The theory of wave propagation has been developed and extended to thermomechanical materials with memory. In many cases waves of sufficiently small amplitude are damped as they progress, but weak waves of sufficiently great amplitude are reinforced in such a way as to become shock waves after a finite length of time. Thus damping, well known from the linearized theories of the last century, and reinforcement, a typical phenomenon of non-linearity, are united and related by the theory. The critical amplitude, which specifies the boundary between damping and reinforcement, is calculated from the mechanical and thermal properties of the material. In a word, we may say that mechanics and thermodynamics are now fused into a single, embracing thermomechanics of deformable bodies. Of those who in recent years have solved or gone far toward solving major problems of mechanics set but left open in the nineteenth or even the eighteenth century should be mentioned, in alphabetical order: Stuart Antman (born 1939), Gaetano Fichera (born 1922), Kurt Otto Friedrichs (born 1901), Harold Grad (born 1923), Daniel Donald Joseph (born 1929), Alek- sandr Yakolevich Khinchin (1894-1959), Mikhail Alekseevich Lavrentiev (born 1900), David Ruelle (born 1935), and James Burton Serrin (born 1926). Among the creators of the new thermomechanics, "classical" in the best sense of the word, may be named Bernard David Coleman (born 1930), Jerald Laverne Ericksen (born 1924), Morton Edward Gurtin (born 1934), Walter Noll (born 1925), Ronald Samuel Rivlin

(born 1915), and Richard A. Toupin (born 1926). Some of these names, notably those of Rivlin, Serrin, and Toupin, should appear on both lists. A different and longer list would be required in order to include those who have done important work in analysis and geometry that has immediate bearing on problems of mechanics.

Retrospect

The history of mechanics shows abundantly that there is no one "scientific method" that should be used in all cases and at all times. Different methods, different "philosophies of science", have been successful at different periods, for different problems, and sometimes for the same man on different occasions. Perhaps after the arid if not blind formal manipulation within the tight compartments of old theories that the latter nineteenth century accepted as being theoretical mechanics, even Prandtl's and Taylor's return to rough, physical guesswork, with all its contempt for "pure" mathematics and its worship of lists of long numbers extracted from costly complexes of pipes and wires, was a healthy reaction-for its day. But reactions provoke reactions. The recent creation of rational thermomechanics illustrates again the approach used most commonly in the eighteenth century and the first half of the nineteenth: induction of general theory from important examples and successful previous theories for special phenomena, calling always upon the full power of mathematics to derive many small truths, sometimes unexpected, from a few large ones. Both directly and indirectly it has drawn upon much earlier work - indeed, upon the history of mechanics - to find inspiration and example as well as to search out forgotten discoveries, the value of which can only now be estimated.

The research reported here was supported by a grant of the U.S. National Science Foundation. I am indebted to J.L. Ericksen for criticism of the first draught of this paper.

Received January 3, 1975


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